1. **Problem Statement:** Find the matrix $A$ representing the linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^3$ defined by $$T\begin{pmatrix}x \\ y\end{pmatrix} = \begin{pmatrix}2x - y \\ x + 2y \\ 3y\end{pmatrix}$$ with respect to the standard bases. 2. **Recall:** The matrix $A$ of a linear transformation $T$ is found by applying $T$ to the standard basis vectors of $\mathbb{R}^2$ and writing the results as columns. 3. **Apply $T$ to the first basis vector:** $$T\begin{pmatrix}1 \\ 0\end{pmatrix} = \begin{pmatrix}2(1) - 0 \\ 1 + 2(0) \\ 3(0)\end{pmatrix} = \begin{pmatrix}2 \\ 1 \\ 0\end{pmatrix}$$ 4. **Apply $T$ to the second basis vector:** $$T\begin{pmatrix}0 \\ 1\end{pmatrix} = \begin{pmatrix}2(0) - 1 \\ 0 + 2(1) \\ 3(1)\end{pmatrix} = \begin{pmatrix}-1 \\ 2 \\ 3\end{pmatrix}$$ 5. **Form the matrix $A$ by placing these images as columns:** $$A = \begin{pmatrix}2 & -1 \\ 1 & 2 \\ 0 & 3\end{pmatrix}$$ **Final answer:** $$A = \begin{pmatrix}2 & -1 \\ 1 & 2 \\ 0 & 3\end{pmatrix}$$